Introduction to symbolization

Why structure matters

Here’s an argument that is clearly valid:

If each sentence got its own letter (A, B, C, D) the symbolic version would show no connection between premises and conclusion at all. Symbolizing the structure instead, with B for “the bakery is open,” C for “the café is open,” and F for “there is fresh bread,” gives:

Now the shared parts are visible, and later chapters give you tools to prove the argument valid.

Symbolization keys

A symbolization key lists which English sentence each sentence letter stands for:

B

The bakery is open.

C

The café is open.

F

There is fresh bread.

• Within one key, each letter stands for exactly one sentence.
• Picking a letter that reminds you of the sentence helps, but it isn’t required. With many sentences on a similar topic, you’ll run out of good reminders.
• A key also lets you read SL back into English. With the key above, ∼(B & C) reads “It is not the case that both the bakery and the café are open.”

Truth-functional connectives

A connective is used truth-functionally when the truth value of the compound it builds is completely settled by the truth values of the sentences it connects, whatever those values are. The connectives of SL are truth-functional by definition. Their characteristic truth tables say everything about them:

• ∼P is true exactly when P is false.
• P & Q is true exactly when both conjuncts are true.
• P ∨ Q is true when at least one disjunct is true (including both).
• P ⊃ Q is false only when the antecedent is true and the consequent is false.
• P ≡ Q is true exactly when both sides have the same truth value.

Want to play with these? The truth table refresher has a connective lab.

The two-step method

1. Paraphrase. Rewrite the English sentence using only the connectives “it is not the case that”, “and”, “or”, “if … then”, and “if and only if”, with complete sentences as the parts. If a sentence can’t be paraphrased as a truth-functional compound, it serves as its own paraphrase.
2. Symbolize. Swap each atomic sentence for its letter and each connective for its symbol. The paraphrase and the SL sentence have the same structure.

For example, “Milo will come if his shift ends early” becomes:

Two changes happened in the paraphrase: the “if” part moved to the front, and “his” became “Milo’s” so each part makes sense on its own.

How these exercises work: some start by asking you to pick the right paraphrase, then you type the symbolization. Any equivalent answer counts. If you’re wrong, you’ll see your sentence read back in English and a case where it comes apart from the original.

Negation

The tilde corresponds to “it is not the case that.” English hides negation in many places: “isn’t”, “not”, and prefixes such as un-, in-, non-.

Careful with “someone” and “no one.” “Someone isn’t ready” denies that everyone is ready. It doesn’t deny that someone is ready. And “No one is ready” denies that someone is ready. Put the “not” in front of the right claim.

Conjunction

The ampersand corresponds to “and.” Many English words assert that two things are both true while adding a hint of contrast or surprise: but, yet, although, however, nevertheless, moreover, while. The hint is lost in the paraphrase, but what’s actually asserted is kept. That’s usually all logic needs.

Sentences like “Lena and Theo both speak Portuguese” aren’t literally two sentences joined by “and”, but they’re equivalent to one that is.

Disjunction

The wedge corresponds to “or”, and it is inclusive: P ∨ Q is true when either or both disjuncts are true. “Unless” also becomes “or”: “The game is Saturday unless it rains” becomes “The game is Saturday or it rains.”

Exclusive “or”. When you really mean “one or the other but not both,” say so explicitly: (P ∨ Q) & ∼(P & Q). Often the “not both” comes from the situation (only one person can win a race), not from the word “or” itself.

Material conditional

The horseshoe corresponds to “if … then.” The hard part is telling the antecedent from the consequent, because English moves them around.

“If” vs. “only if”. “If” introduces the antecedent. “Only if” introduces the consequent. “You’ll graduate only if you pass the language exam” says passing is required, not that passing guarantees graduation. So it’s G ⊃ P, not P ⊃ G.

Material biconditional

The triple bar corresponds to “if and only if.” Variants include “if but only if” and sometimes “just in case.”

“Just in case” has two uses. “The alarm sounds just in case the door opens” gives an exact condition: biconditional. “I brought snacks just in case we get hungry” explains a precaution: not a biconditional.

Review

First a quick round on spotting the right connective, then every 2.2 symbolization mixed together.